Take the expression: $$\iiint_V \nabla \cdot \textbf{u}\ dV$$
It seems intuitive that we should be able to go from this expression and express $$V$$, $$\textbf{u}$$, $$\nabla$$, and $$dV$$ in whatever coordinate system we want at this point, but plugging the spherical coordinates straight into the expression yields the wrong answer - one has to add the Jacobian just as if the first step was expressing everything in Cartesians and converting to polars. Why do Cartesians get such special treatment? My expectation would be because the eigenvectors are constant throughout space, but I would appreciate a more thorough explanation.